DO NOT USE A CALCULATOR IN THIS QUESTION. Write \(\quad(5-\sqrt{3})(\sqrt{6}+\sqrt{2})^{-2}\) in the form \(a+b \sqrt{3}\), where \(a\) and \(b\) are constants.

DO NOT USE A CALCULATOR IN THIS QUESTION. Write \(\frac{16+11 \sqrt{10}}{2+\sqrt{10}}+1\) in the form \(p+q \sqrt{10}\), where \(p\) and \(q\) are integers.

Write

\(\frac{\sqrt{9p^2q}\times r^{-3}}{(2p)^3q^{-1}\sqrt[5]{r}}\)

in the form \(kp^aq^br^c\), where \(k\), \(a\), \(b\) and \(c\) are constants.

Do not use a calculator in this question.

Write

\(\frac{4-\sqrt5}{7-3\sqrt5}\)

with a rational denominator, simplifying your answer.

Do not use a calculator in this question.

(a) Simplify

\(\frac{\sqrt{128}}{\sqrt{72}}.\)

(b) Simplify

\(\frac{1}{1+\sqrt3}-\frac{\sqrt3}{3+2\sqrt3},\)

giving your answer as a fraction with an integer denominator.

Without using a calculator, express \(\dfrac{(\sqrt5-3)^2}{\sqrt5+1}\) in the form \(p\sqrt5+q\), where \(p\) and \(q\) are integers.

(a) Simplify \(\dfrac{(3+2\sqrt5)(6-2\sqrt5)}{4-\sqrt5}\), giving your answer in the form \(a+b\sqrt5\), where \(a\) and \(b\) are integers.

(b) Triangle \(ABC\) is such that \(AB=6-2\sqrt3\), \(BC=6+2\sqrt3\), and \(\cos ABC=-\dfrac12\). Find \(AC\), giving your answer in the form \(c\sqrt d\), where \(c\) and \(d\) are integers.

Do not use a calculator in this question.

(a) Simplify

\(\frac{5+6\sqrt5}{6+\sqrt5}.\)

(b) Show that

\(3^{0.5}\times(\sqrt2)^7\)

can be written in the form \(a\sqrt b\), where \(a\) and \(b\) are integers and \(a\gt b\).

(c) Solve the equation

\(x+\sqrt2=\frac{4}{x},\)

giving your answers in simplest surd form.

Do not use a calculator in this question.

(a) Simplify

\(\frac{5+6\sqrt5}{6+\sqrt5}.\)

(b) Show that

\(3^{0.5}\times(\sqrt2)^7\)

can be written in the form \(a\sqrt b\), where \(a\) and \(b\) are integers and \(a\gt b\).

(c) Solve the equation

\(x+\sqrt2=\frac{4}{x},\)

giving your answers in simplest surd form.

Do not use a calculator in this question.

(a) Simplify

\((\sqrt2+2\sqrt5)(4\sqrt2-3\sqrt5),\)

giving your answer in the form \(a+b\sqrt c\), where \(a\), \(b\) and \(c\) are integers.

(b) Simplify

\(\frac{4-3\sqrt6}{\sqrt3+\sqrt2},\)

giving your answer in the form \(p\sqrt3+q\sqrt2\), where \(p\) and \(q\) are integers.

Do not use a calculator in this question.

(a) Show that

\(\sqrt{24}\times\sqrt{27}+\frac{9\sqrt{30}}{\sqrt{15}}\)

can be written in the form \(a\sqrt2\), where \(a\) is an integer.

(b) Solve the equation \(\sqrt3(1+x)=2(x-3)\), giving your answer in the form \(b+c\sqrt3\), where \(b\) and \(c\) are integers.

Without using a calculator, express

\(\left(\frac{1+\sqrt5}{3-\sqrt5}\right)^{-2}\)

in the form \(a+b\sqrt5\), where \(a\) and \(b\) are integers.

Given that \(z=a+(a+3)\sqrt3\) and \(z^2=79+b\sqrt3\), find the value of each of the integers \(a\) and \(b\).

If \(z=2+\sqrt3\), find the integers \(a\) and \(b\) such that \(az^2+bz=1+\sqrt3\).

Find integers \(p\) and \(q\) such that

\(\dfrac{p}{\sqrt3-1}+\dfrac{1}{\sqrt3+1}=q+3\sqrt3.\)